Angel "Java" Lopez en Blog

Publicado el 5 de Junio, 2013, 6:50

Anterior Post
Siguiente Post

Hacía un tiempo que no escribía sobre el tema. Esta es una serie de notas personales, más que un estudio ordenado del tema. Son notas para servir de base. Las paso por escrito para no perderlas.

Nota 2

Ver Feynman Diagrams for the Masses (part 1) del bueno de Carl Brennan. Leo ahí:

In the usual way of doing physics, one obtains Feynman diagrams after making a guess at the Lagrangian density. Joseph Louis Lagrange was an 18th century mathematician. The Lagrangian is roughly the kinetic energy minus the potential energy. If we choose a particular form for the kinetic and potential energies we can write down the Lagrangian. From the Lagrangian we can compute the equations of motion. We do this by varying the Lagrangian, that is, by computing the Lagrangian for a set of possible paths and picking a path for which small changes to the path do not change the Lagrangian. Such a path is a possible sequence of values for the positions of our particles (and their momenta). The equations of motion will show up as a set of coupled differential equations.

For a wave theory, like quantum mechanics, the kinetic and potential energies are defined at each point in space-time as a functional of the fields. With psi  the wave function, T the kinetic energy, and V the potential energy, one could write the Lagrangian as: 

También describe lo que es un propagador, funciones de Green, relación con Fourier, etc... Leo:

In the usual way of doing physics, one obtains Feynman diagrams after making a guess at the Lagrangian density. Joseph Louis Lagrange was an 18th century mathematician. The Lagrangian is roughly the kinetic energy minus the potential energy. If we choose a particular form for the kinetic and potential energies we can write down the Lagrangian. From the Lagrangian we can compute the equations of motion. We do this by varying the Lagrangian, that is, by computing the Lagrangian for a set of possible paths and picking a path for which small changes to the path do not change the Lagrangian. Such a path is a possible sequence of values for the positions of our particles (and their momenta). The equations of motion will show up as a set of coupled differential equations.

For a wave theory, like quantum mechanics, the kinetic and potential energies are defined at each point in space-time as a functional of the fields. With psi  the wave function, T the kinetic energy, and V the potential energy, one could write the Lagrangian as: 

L(t) = int_{z=-infty}^{+infty}int_{y=-infty}^{+infty}int_{x=-infty}^{+infty}( T(psi(x,y,z,t)) - V(psi(x,y,z,t))dx;dy;dz

. Instead of getting an equation of motion for the particles, we get a set of coupled partial differential equations. The partial derivatives show up because of the dependency on position.

If we turned off the interaction, the equations of motion we would get from the Lagrangian in the usual QFT technique would be something like Schrödinger"s equation or Dirac"s equation. The propagators (Green"s functions) for these equations of motion are well known. What is not known are the propagators for the more complicated equations of motion that would give the full Lagrangian. Such a propagator is called "exact". We will direct our effort at this sort of problem, that is, finding the exact propagators (or an approximation to them) for complicated Lagrangians.

Y sigue con la discusión ahí.

Sobre Carl Brennan:

Carl Brannen is well known to the regulars of this blog. He is an independent researcher and my favourite non-professional theorist, because he gives me the hope that brilliant minds, who were diverted from the natural path of doing basic research, may return to it for good. And Carl provides us with another important proof: that institutionalized science does sometimes listen to the voice of those who have something to say regardless of who signs their monthly paycheck. It may give them a hard time getting their papers published, though.

Lo encuentro citado en Guest Post: Carl Brannen, "Position, Spin, And The Particle Generations"

Nota 3

Encuentro lagrangianos en el artículo de Gauge Theory de la Wikipedia. Por ejemplo:

An example: Scalar O(n) gauge theory [edit]

The remainder of this section requires some familiarity with classical or quantum field theory, and the use of Lagrangians.
Definitions in this section: gauge group, gauge field, interaction Lagrangian, gauge boson.

The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between fields which were originally non-interacting.

Trata el tema de lagrangiano con simetría global, y luego pasa a local:

Now, demanding that this Lagrangian should have local O(n)-invariance requires that the G matrices (which were earlier constant) should be allowed to become functions of the space-time coordinates x.

Unfortunately, the G matrices do not "pass through" the derivatives,....

Más adelante:

The difference between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian

 mathcal{L}_mathrm{int} = rac{g}{2} Phi^T A_{mu}^T partial^mu Phi + rac{g}{2}  (partial_mu Phi)^T A^{mu} Phi + rac{g^2}{2} (A_mu Phi)^T A^mu Phi

This term introduces interactions between the n scalar fields just as a consequence of the demand for local gauge invariance. 

Ver Interaction Lagrangian. Y por supuesto, tenía que llegar a estas nodas: Lagrangian.

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was introduced in a reformulation of classical mechanics introduced by Joseph Louis Lagrangein 1788, known as Lagrangian mechanics.

Y ver Hamiltonian mechanics:

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of thequantum mechanics.

Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting fromLagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788.

Como ven, son temas que nos van llevando por todos lados. De ahí, que tenga que escribir estas notas ;-).

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Por ajlopez, en: Ciencia